Simplify and expand the following expression: $ \dfrac{a - 2}{3a + 3}-\dfrac{4a}{2a + 2} $
Explanation: In order to subtract expressions, they must have a common denominator. Get both fractions over a common denominator of $(3a + 3)(2a + 2)$ Multiply the first term by $\dfrac{2a + 2}{2a + 2}$ $ \begin{align*} \dfrac{a - 2}{3a + 3} \times \dfrac{2a + 2}{2a + 2} & = \dfrac{(a - 2)(2a + 2)}{(3a + 3)(2a + 2)} \\ & = \dfrac{2a^2 - 2a - 4}{(3a + 3)(2a + 2)}\end{align*} $ Multiply the second term by $\dfrac{3a + 3}{3a + 3}$ $ \begin{align*} \dfrac{4a}{2a + 2} \times \dfrac{3a + 3}{3a + 3} & = \dfrac{(4a)(3a + 3)}{(2a + 2)(3a + 3)} \\ & = \dfrac{12a^2 + 12a}{(2a + 2)(3a + 3)}\end{align*} $ Now we have: $ = \dfrac{2a^2 - 2a - 4}{(3a + 3)(2a + 2)} - \dfrac{12a^2 + 12a}{(2a + 2)(3a + 3)} $ Now both terms have a common denominator we can subtract the numerators: $ = \dfrac{2a^2 - 2a - 4 - (12a^2 + 12a)}{(3a + 3)(2a + 2)} $ $ = \dfrac{2a^2 - 2a - 4 - 12a^2 - 12a}{(3a + 3)(2a + 2)} $ $ = \dfrac{-10a^2 - 14a - 4}{(3a + 3)(2a + 2)}$ Expand the denominator: $ = \dfrac{-10a^2 - 14a - 4}{6a^2 + 12a + 6}$ Simplify: $ = \dfrac{-5a^2 - 7a - 2}{3a^2 + 6a + 3}$